If $dI_n= AB $, must $\det (A) $ divide $d $?

Question

Let $A,B$ be matrices with integer coefficents. Let $d I_n $ be the $n $ by $n $ identity matrix with the one entries replaced by $d $ entries. ($d$ is an integer). If $d I_n= AB $ , must $\det (A) $ divide $d $ ?

If the answer is yes, does this hold for all commutative rings in general?

Thank you,

Answer 1

Try $$A = \pmatrix{2 & 0\cr 0 & 2\cr},\ B = \pmatrix{1 & 0\cr 0 & 1\cr},\ d = 2$$

In general we have $B = (d/\det(A)) \text{adj}(A)$, where $\text{adj}(A)$ is the [classical adjoint] of $A$ is also an integer matrix. We have a counterexample whenever the entries of $\text{adj}(A)$ are all divisible by some factor of $\det(A)$.